From the famous Noether's theorem, we know that symmetries and conserved quantities are two sides of the same coin. In fact, this theorem states that we can associate a conserved quantity (conserved charge) with every continuous global symmetry. A natural question arises: can we also associate conserved charges with local continuous symmetries (such as those found in electromagnetic and gravitational theories)? The answer is that a certain class of these symmetries can indeed be associated with conserved charges. One method of associating charges with these symmetries is through the method of covariant phase space (although this method also works for global symmetries). In essence, this method can be considered a generalization of canonical mechanics (the principle of least action) constructed in a covariant way (independent of the choice of time orientation). In this talk, we will first provide a relatively simple overview of this method. Then, considering the fluctuations at the boundaries, we will extend this method. Finally, we will briefly discuss the compatibility of this approach with the first law of thermodynamics for black holes and the Gibbs-Duhem relation.
